2 Ulmke et al. been studied in a variety of systems. Doping with static scatterers like nonmagnetic impurities usually weakens antiferromagnetic order

Transcription

1 Disorder and Impurities in Hubbard-Antiferromagnets M. Ulmke 1, P. J. H. Denteneer 2, V. Janis 3, R. T. Scalettar 4, A. Singh 1;, D. Vollhardt 1, G. T. Zimanyi 4 1 Theoretische Physik III,Universitat Augsburg, Augsburg, Germany 2 Lorentz Institute for Theoretical Physics, University of Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 3 Institute of Physics, Academy of Science of the Czech Republik, CZ Praha 8, Czech Republik 4 Department of Physics, University of California, Davis, CA 95616, USA Summary: We study the inuence of disorder and randomly distributed impurities on the properties of band-antiferromagnets. To this end the Hubbard model with (i) random potentials, (ii) random hopping elements, and (iii) randomly distributed values of interaction is treated using quantum Monte Carlo and dynamical mean-eld theory. In cases (i) and (iii) weak disorder can lead to an enhancement of antiferromagnetic order: in case (i) by a disorder-induced delocalization, in case (ii) by binding of free carriers at the impurities. For strong disorder or large impurity concentration antiferromagnetism is eventually destroyed. Random hopping leaves the local moment stable but AF order is suppressed by local singlet formation. Random potentials induce impurity states within the charge gap until it eventually closes. Impurities with weak interaction values shift the Hubbard gap to a density o half-lling. In both cases an antiferromagnetic phase without charge gap is observed. 1 Introduction Antiferromagnetic spin correlations are present in many strongly correlated electron systems, notably the prototype Mott insulators NiO and V 2 O 3, the parent compounds of HTSC cuprates, and heavy fermion systems such as YbP, U 2 Zn 17, and many others. Many of those systems are intrinsically disordered, in particular upon additional homo- or heterovalent doping. The inuence of impurity doping on antiferromagnetic (AF) order and electronic properties has recently

2 2 Ulmke et al. been studied in a variety of systems. Doping with static scatterers like nonmagnetic impurities usually weakens antiferromagnetic order, a prominent example being Zn doping in YBa 2 Cu 3 O 6 [1]. In spin chains (CuGeO 3 ) [2] and ladder compounds (SrCu 2 O3) [3] on the other hand, doping with magnetic and nonmagnetic impurities can induce AF order while the pure systems show spin gap behavior. Very eective in destroying AF order are mobile carriers, e.g. hole doping in La 1 x Sr x CuO 4 [4]. The stability of AF order strongly depends on the positions of the dopand level. While in the nickel oxides La 1 x Sr x NiO 4 [5] and Ni 1 x Li x O [6] holes are supposed to be localized, in the cuprate La 1 x Sr x CuO 4 the hole level lies in the valence band leading to mobile scatterers. As a result, AF order is stable in La 1 x Sr x NiO 4 up to x = 0:5, but is destroyed in the cuprate already at approx. 5% Sr doping. In the present paper we will study the inuence of disorder on AF order and the Mott band gap in band-antiferromagnets. We employ the Hubbard model in the presence of dierent types of disorder. While the (disordered) Hubbard model is certainly far too simple to describe real materials it already contains very rich physics including local moment formation, magnetic ordering, Mott- Hubbard transition, and Anderson localization. On the other hand, the interplay of disorder and interactions in electronic systems belongs to the most dicult problems in physics, and reliable results within simple models are still very desirable. The problem has been investigated in the past by a variety of methods, including eld theoretical approaches [7], renormalization group treatments [8, 9], unrestricted Hartree-Fock [10, 11], dynamical mean-eld theory (DMFT) [12, 13, 14], quantum Monte Carlo (QMC) [15, 16, 17], and several more (see [18] for a review). Here we give an overview of results obtained by QMC and DMFT concentrating on the AF phase diagram and the Mott gap. We consider the following Hubbard Hamiltonian: X X X ^H = ( i )^n i + t ij (^c y i^c j + h:c:) + U i (^n i" 1 2 )(^n i# 1 ): (1.1) 2 i i hiji In principle all parameters i ; t ij ; U i can be randomly distributed. The precise denition of the dierent disorder types studied in this paper will be given in the following sections. The average t av ht ij i sets our energy scale. We will restrict the hopping t ij to nearest-neighbors hence not allowing for frustration. Longer range, random hopping amplitudes will be important in the modeling of amorphous materials such as doped semiconductors [18, 19] and are not considered in the present work.

3 Disorder and Impurities in Hubbard-Antiferromagnets 3 2 Methods 2.1 Determinant quantum Monte Carlo (d=2) We use a nite temperature quantum Monte Carlo method to obtain approximation-free results for nite lattices [20]. The algorithm is based on a mapping of the interacting electron problem onto a d+1 dimensional quasi-classical problem using auxiliary Ising-type spins. It provides for calculating thermal averages of observables, ^A, h ^Ai = Tr ^A e ^H Tr e ^H : (2.2) The phase space sampling over the auxiliary eld congurations is performed using Monte Carlo techniques. The weight of a conguration is proportional to a product of two determinants, one for each electron spin species. In the case of half-lling without random potentials, i.e. i 0, on a bipartite lattice the determinants always have the same sign, hence their product is always positive semi-denite, which can be shown by particle-hole transformation of one spin species [c i#! ( 1) i c y i# ]. In the general situation that the product can become negative the algorithm is still applicable in principle, however the signal to noise ratio decreases exponentially with systems size, inverse temperature and interaction, putting severe restrictions to the applicability of the method. This so-called \minus-sign problem" is a general obstacle for all exact fermionic Monte Carlo methods as well as for spin-systems in the presence of frustration. Even without the minus-sign problem the computational eort is large because the computer time grows cubically with system size N, restricting N to the order of 100 on present supercomputers. In the case of disorder all observables have to be averaged over the (frozen) disorder congurations. Because of the computational eort we restrict ourselves to two dimensional lattices with linear size up to L x = 10 lattice spacing, which often allows a reliable nite size scaling. Since we are interested in AF ordering we calculate the magnetic correlation functions C(l) and their Fourier transforms, the magnetic structure factors S(q), C(l) = 1 N X j hm j m j+li; S(q) = X l C(l)e iql : (2.3) In particular the AF structure factor S(; ) is used to obtain the ground state sublattice magnetization M by a nite size scaling Ansatz according to spinwave theory [21]: S(; ) N = M O( 1 L x ): (2.4)

4 4 Ulmke et al. 2.2 Dynamical mean-eld theory (limit of innite dimensions) The dynamical mean-eld theory [22, 23, 24] is a local approximation in which the self energy becomes site diagonal, or momentum independent: ij (!) = ij (!); ~(k;!) = ~(!): (2.5) The one-particle Green function G(k;!) can hence be obtained from the noninteraction Green function G 0 (k;!) by G(k;!) P = G 0 (k;! ~(!)), and the local Green function is given by G ii (!) = 1=N G(k;!) This does not imply k a simple shift of energies, like in traditional mean-eld theories (e.g. Hartree- Fock), because remains dynamical, i.e. frequency depending, preserving local quantum uctuations. The local approximation becomes exact in the limit of innite spatial dimensionality and maps the interacting lattice model onto a self-consistent single impurity model like, for example, the Wol model [25]: ^H Wol = X k " k^n k + U(^n 0" 1 2 )(^n 0# 1 2 ) + (^n 0" ^n 0# ): (2.6) Here the one-particle energies " k have to be dened such that the non-interacting local Green function of the Wol model fullls G 0 Wol (G ii ~) 1. In the self-consistent solution the local (interacting) Green function of the Wol model has to be equal to G ii. In the presence of disorder one has to average over all possible values of U or, respectively. This type of local averaging is equivalent to the \coherent potential approximation", well known from investigations of disordered alloys. While the self-consistency is rather easily reached by iteration the solution of the single impurity problem is the hard part. There exists no analytic solution and dierent numerical and approximative techniques have been employed (see, e.g., [23, 24]). Here we again use auxiliary eld QMC [26], an algorithm quite similar to the one for nite dimensional lattices sketched above. The computer time grows like L 3 where L is the number of Matsubara frequencies. Fortunately, QMC for the single band model is free from the minus-sign problem. Typical quantities under consideration are the staggered magnetic susceptibility AF whose divergence signals the transition to an AF ordered state. Within the ordered phase one can calculate spin-dependent local densities n on a given sublattice and hence the staggered moment M: n = 1 + T X n G n ; M = X n : (2.7) with G n being the local Green function at Matsubara frequency i! n = (2n + 1)T. The one-particle density of states (DOS) is obtained by analytical continuation of the imaginary time Green function using the Maximum Entropy method. For details of the algorithm, the implementation of disorder averages and determination of expectation values see [23, 24, 14].

5 Disorder and Impurities in Hubbard-Antiferromagnets 5 3 Random potentials 3.1 Local moment quenching Random potentials are the most frequently studied type of disorder in the context of Anderson localization. Contrary to the Hubbard interaction which at half-lling favors single occupation on each site, dierent local potentials lead to dierent occupations and hence to a quenching of local magnetic moments on sites with large absolute value of the local potential. This is seen in Fig. 1a where the average local moment squared, m 2, is plotted versus disorder strength for a at distribution of values with width. It is also shown that the spinspin-correlations go in parallel with m 2. For a large width of random potentials this type of disorder is apparently very eective in the destruction of magnetic order. To study the behavior of the charge gap, the electronic is calculated as a function of (Fig. 1b). While disorder decreases in the non-interacting case, is enhanced by disorder at nite U. The reason is the introduction of states within the AF charge gap as will be discussed below. As mentioned above, random potentials break the particle-hole symmetry (a) (b) Figure 1 (a) Local magnetic moment C(0; 0), spin-spin correlation function at lattice vector (2; 2), and AF structure factor S(; ) as a function of disorder strength for a constant distribution of random potentials (in all cases the U = = 0 values are subtracted). (b) compressibility vs. ; disorder reduces in the non-interacting case and enhances it at U = 4 [16].

6 density 6 Ulmke et al. leading to a minus-sign problem even at half lling. This is the reason why only small lattices (4 4 in Fig. 1) can be studied for this type of disorder. 3.2 Disorder-enhanced delocalization Consider the situation deep in the AF phase, i.e. with a large staggered moment. A high value of the local potential on a given site reduces the local spin leading to an eective delocalization of the electron. A low potential on the other hand cannot signicantly further increase localization because the majority spin is already almost saturated. This asymmetry of the localizing and delocalizing eects of random potentials on the majority spins is depicted in the DMFT results, Fig. 2. The majority spin (") is strongly reduced for > 0 but almost unchanged for < 0. Note that the total local density monotonically decreases with, and that the net magnetization (n " n # ) is reduced for any jj > U = 8; W = 8; = 7:5 n " n " # n # " + # i Figure 2 Local spin resolved densities in DMFT as a function of the random potential value i [27]. Total width of disorder distribution is W = 8. The fact that the delocalization is not compensated can be expressed in terms of an enhanced eective hopping parameter t e. In the case (U ) t the eective t e can be estimated by the mapping onto the AF Heisenberg model: the AF exchange arises from virtual hopping of an electron with spin to a neighboring site occupied with an electron of spin. The exchange energy for this and the opposite process is J ij = t 2 U ( i j ) + t 2 U ( j i ) 2t2 U 1 + ( i j ) 2 U 2 : (3.8)

7 Disorder and Impurities in Hubbard-Antiferromagnets 7 Figure 3 AF (T n U) phase diagram in DMFT for a bimodal random potential distribution. Energies in unit of the half bandwidth. [14] J ij is hence always enhanced from the pure case (J 0 = 2t 2 =U), and the average exchange becomes J av = J 0 [1 + (=U) 2 ] where depends only on the shape of the disorder distribution (e.g., = 1=12 for the constant, = 1=2 for a binary distribution). More generally, one can show that the form of the magnon propagator remains unchanged in second order in =U [27], but magnons are stiened by a factor of (1 + (=U) 2 ). The eective hopping in strong coupling can thus be expressed as t e = t[1 + (=U) 2 ] 1=2. In dimensions d > 2 one expects the Neel temperature T N to be proportional to J and hence T N, too, should be enhanced by weak disorder. While disorder-enhanced delocalization stabilizes AF order at strong coupling it suppresses it at weak coupling. For small U the AF state is rather \spin density wave like" and an enhanced kinetic energy tends to weaken AF order and to reduce T N. In addition, random potentials destroy the perfect nesting instability responsible for AF order at small U. The T vs. U phase diagram within DMFT (Fig. 3) summarizes and conrms the above considerations: For small U, T N is reduced and eventually vanishes when becomes roughly equal to U. At large U, however, the T N curves for dierent cross each other, i.e. at a given value of U, T N increases with disorder. The opposite eects of disorder on T N depending on U are due to the non-monotic behavior of the function T N (U). If the slope of T N (U) is positive (negative) an eectively reduced ratio U=t e leads to a suppression (enhancement) of T N.

8 8 Ulmke et al. 3.3 Closing of the charge gap Local potentials can induce electronic states in the charge gap. As also found experimentally the positions of the impurity states are crucial for the stability of AF order with respect to carrier doping. Disorder-induced gap states will reduce the size of the charge gap. We now want to determine the critical disorder strength at which the gap vanishes. First we employ the T-matrix approximation which becomes exact for a single impurity and will be used as an approximation in the disordered case. [28, 27] To apply this approach we would need the oneparticle Green function of the correlated, not-disordered problem which is of course still unknown. As an approximation we use the local host Green function of the AF Hartree-Fock solution: [g 0 ] ii = (1=N) X k (! D)=(! 2 E 2 k): (3.9) p Here 2D = mu is the Hubbard energy gap in the pure AF, E k = D k is the P AF band energy, and D is obtained from the self-consistency condition (1=N) (2E k k) 1 = U 1. With this local host Green function we can calculate the location of impurity-induced states from the poles in the T-matrix, T (!) = i =(1 i [g(!)] 0 ii ). We consider a constant distribution of random potential between =2. There are disorder-induced states within the gap if j[g] 0 ii j > 2=. The energy D ~ up to which states are formed within the gap is given by [g( 0 D)]ii ~ = 2=. The remaining charge gap 2 D ~ is plotted versus in Fig. 4. The decrease is almost linear and the 2 D ~ vanishes close to = U. Also shown is the result from a numerical unrestricted Hartree-Fock (UHF) analysis. In this approach, the HF Hamiltonian on a nite lattice is numerically (self-consistently) diagonalized, so that disorder is treated exactly [10, 11, 27]. The energy gap is obtained from the energy dierence between the lowest energy state of the upper Hubbard band and the highest energy state of the lower Hubbard band. Averages are taking over 100 disorder realizations on a lattice. The saturation of the gap at =U 1 is due to the nite system size. The agreement with the T- matrix approach is excellent for the present interaction value U = 10t. Deviations from the T-matrix approach are more pronounced at lower interaction strengths where the fermion states are more extended. The closing of the charge gap is also observed within the DMFT approach. Fig. 5a shows the density of states for several disorder values. Note that all spectra shown are within the AF ordered phase, i.e. AF order is much more stable than the charge gap which closes about = U. A linear reduction of the charge gap is also observed in the case of a binary disorder distribution (Fig. 5b). For this stronger type of disorder both the charge gap and the AF order vanish near = U.

9 Normalized Energy Gap Disorder and Impurities in Hubbard-Antiferromagnets U = W=U Figure 4 Normalized energy gap, D=D, ~ in d = 2 versus disorder strength (W ) at U = 10t as obtained by the T-matrix approximation (solid line) and the UHF analysis (symbols) [27]. N(ω) U=8, β= W Charge gap U=10.6, T=0.125 binary distribution (a) 8 ω (b) Figure 5 (a) Density of states (DOS) for a constant disorder distribution of dierent width in DMFT [27]; all spectra are within the AF phase. (b) Charge gap vs. in DMFT for the bimodal distribution of random potentials. 3.4 Crossover to strong disorder; spin vacancies As discussed in the previous section the one-particle gap decreases with and vanishes near = U. Upon further increase of the two Hubbard bands will overlap whereby electrons from the highest levels of the lower band (with i > U=2) will be transfered to the lowest levels of the upper band (with i < U=2). Those sites will become doubly occupied, their local moment will vanish as observed in Sec Assuming at Hubbard bands of width, appro-

10 10 Ulmke et al. (a) (b) Figure 6 (a) T N as a function of site dilution in DMFT for dierent values of U. Dashed line Hartree-Fock approximation for U = 0:65 (energies in units of the half bandwidth). (b) Enhancement of local moments on the remaining fraction (1 x) of the lattice at inverse temperature = 16. [14] priate in the limit t U;, the overlap region is ( U) and the fraction of nonmagnetic site can be estimated by x ( U)=. This situation can be modeled by spin vacancies of concentration x which will lead to strong magnon scattering. Such diluted antiferromagnets have been widely studied recently mostly within spin diluted Heisenberg models [29, 30] and also Hubbard models [28]. A perturbative analysis in the strong coupling limit (O(t 2 =U 2 )) [28, 27] yields a softening of magnons by a factor of (1 2x) in d = 2. Extrapolating to large fractions x, the magnon energy scale ~ J = J(1 2x) hence vanishes only close to the percolation threshold x perc 0:4. In d > 2 the Neel temperature is assumed to be proportional to ~ J and a linear decrease of TN is thus expected. In experiments, e.g. on Zn doped YBa 2 Cu 3 O 6 [1] and on Li doped NiO [31] such a linear dependence on x is indeed observed with prefactors of 2 and 2.2, respectively. In the limit d! 1 the percolation threshold is 1.0; hence we can expect AF order to persist for arbitrarily large dilution. Within DMFT indeed a linear decrease of T N / (1 x) is observed for the diluted model at large U (Fig. 6a). At small U, however, AF order is remarkably robust, T N being constant up to dilutions close to x = 1 where T N eventually drops to zero. This behavior of T N can be explained by a strong enhancement of the local moment density on the remaining sites at small U (Fig. 6b). The reason for this enhancement is that with reduced average number of nearest neighbors the kinetic energy decreases, leading to a stronger localization. At large U the local moments are already almost saturated at x = 0 and just cannot be further enhanced. The situation

11 Disorder and Impurities in Hubbard-Antiferromagnets 11 in the case of vacancies is therefore quite dierent from the eect of weak site disorder where disorder-enhanced delocalization is observed. 4 Random hopping The case of spin vacancies discussed in the previous section can of course be regarded as a specic type of randomness in the hopping elements. The more generic case of a continuous (at) distribution of t ij 2 [1 =2; 1 + =2] was studied in d = 2 using QMC. Since the hopping is still restricted between nearest neighbors on a square lattice particle-hole symmetry is preserved and no minussign problem occurs at n = 1. Random t ij hardly aect the density of local moments (Fig. 7a), qthe slight decrease may be due to the enhanced kinetic energy which is / < t 2 ij >. Nevertheless, longer-range spin-spin correlations are strongly suppressed if is of the order of t (see Fig. 7a). The nite size scaling according to (2.4) yields the AF order parameter (staggered moment) M versus (Fig. 7b). M vanishes at a critical disorder strength of c 1:4. We propose [16] that the phase boundary is determined basically by the variance of the AF exchange coupling v = (< J 2 ij > < J ij > 2 )= < J 2 ij >. AF order persists for v < v c 0:4. This criterion is consistent with the phase boundary of the bond-disordered AF Heisenberg model with a bimodal distribution of J ij [32]. The reason for the vanishing of AF order for this type of disorder is supposedly the formation of local singlets. Such singlets will form rst on the strongest bonds and will leave some spins which are weakly coupled to their neighbors unpaired. Those \free" spins are expected to give a Curie-like contribution to the susceptibility as observed in doped semiconductor [18, 19]. The numerical results indeed show a strong enhancement of the uniform susceptibility however do not allow for quantitative conclusions [16]. 5 Impurities with weak local interactions The Hubbard model at n = 1 exhibits both AF order and the Mott-Hubbard metal-insulator transition (MIT). It is important to note that both eects are in principle independent. The Mott-Hubbard MIT occurs at intermediate interaction (U bandwidth) while AF order can set in at arbitrarily small U due to the Fermi surface nesting instability. It is argued that the MIT at T = 0 represents a quantum critical point which is however concealed by the low temperature AF phase [33], and attempts are made to suppress the AF phase to very low or zero T by dierent types of frustration [24]. A dierent possibility to separate AF

12 12 Ulmke et al. (a) (b) Figure 7 (a) Local moment and spin-spin correlations in d = 2 for random hopping (same quantities as in Fig. 1). (b) Staggered magnetization M as obtained by nite size scaling vs. [16]. order and the MIT is to shift the lling at which the Mott-Hubbard MIT occurs away from n = 1 by introducing impurities with a low (or zero) local interaction with a concentration f. Such impurity sites can be doubly occupied without the cost of the local repulsion and hence the MIT is expected at a density n = 1 + f. 5.1 Mott-Hubbard gap The Hubbard model with a bimodal distribution of U values (U = 0 on a fraction f of the lattice, and U = 8t on the remaining sites) was investigated by QMC on square lattices and in DMFT. Note that for technical reasons (minus-sign problem) the choice of the U disorder in model (1.1) preserves particle-hole symmetry. This corresponds to dierent chemical potential on the two constituents such that at n = 1 the electronic density is homogeneous, i.e. independent on U i. To detect charge gaps the electronic density n is plotted versus the chemical potential (Fig. 8a). As expected the gap moves to a density o half lling close to 1 + n. This agrees with the kinetic energy which shows minima at the corresponding densities (Fig. 8b). At a closer look one observes an additional gap at n = 1 which is due to the doubling of the unit cell in the AF ordered state.

13 Disorder and Impurities in Hubbard-Antiferromagnets U=8, T=1/8, d=2 n 1.1 f=0.111 f=0.056 f= µ (a) (b) Figure 8 (a) Electronic density n vs. chemical potential for dierent values of U = 0 impurity concentration f. n() has plateaux at n = 1 and close to n = 1 + f. (b) Kinetic energy hki vs. average density for f = 0 and f = 0:111. hki shows minima at n = 1 + f [17]. The gaps at n = 1 and n = 1 + f can also be detected by the one-particle spectrum, obtained in DMFT (see [17]). Between densities 1 and 1 + f the systems is apparently compressible. 5.2 Antiferromagnetic order Figure 9a shows the staggered moment M extrapolated to the thermodynamic limit in d = 2 at n = 1 as a function of impurity concentration f. As expected from the behavior of the charge gap AF order is robust for small f, and vanishes at f c 0:45, i.e. close to the percolation threshold f perc 0:4. O half-lling the nite size scaling is no longer possible due to the minus-sign problem and the AF phase boundary is obtained within DMFT only (shown in Fig. 9b at T = t=8). It is found that the U = 0 impurities can induce AF order at densities for which the clean model is disordered. In the clean model (f = 0) additional electrons are free to move and hence very eective in destroying long range order. While in d = 2 the critical doping is supposed to be zero there is a nite critical doping within DMFT. U = 0 sites provide localizing centers which are energetically favorable for the additional carriers. This is why the AF phase extends to larger dopings at nite (but small) values of f. Since the mechanism of localizing mobile dopands is observed in d = 2, too, we expect the enlargement of the AF phase

14 14 Ulmke et al. T=t/8, U=8t (a) f D AF n (b) Figure 9 (a) Staggered magnetization in d = 2 as a function of impurity concentration f. (b) f n phase diagram at temperature T = 1=8 within DMFT. For small f the AF phase is stabilized against doping. [17] to be present in d = 2 as well. Eventually at larger fractions f the AF phase shrinks, and the critical density approaches 1.0 for f c 0:6 (f c 0:75 in the ground state). For even larger values of f there is no AF order even at n = 1. Both the stabilization of AF order and the shift of the Mott gap to higher densities result from the localization of carriers at the U = 0 impurities. AF order is only destroyed when the density on the U = 0 sites saturates, i.e. it is stable across the Mott-Hubbard MIT. Hence the separation of Mott-Hubbard MIT and AF order is not present in the U = 0 impurity model, at least not for the present parameter values. 6 Summary and Conclusion In this paper we discussed dierent types of disorder in band-antiferromagnets and presented results obtained mostly by quantum Monte Carlo simulations in d = 2 and within dynamical mean-eld theory (d = 1). Dierent mechanism were identied by which disorder can enhance antiferromagnetic order: i) disorder-enhanced delocalization in the case of weakly disordered chemical potentials and ii) localization of doped carriers in the case of impurities with weak local interaction. In both cases compressible antiferromagnetic phases are observed. To determine if the gapless AF phase is metallic

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